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Easy count (Posted on 2014-07-29) Difficulty: 3 of 5
Let S be a set of some positive integers.
We'll call S autonomous if the number of elements in S is itself an element of S. e.g. the set {2,3,5} is autonomous, as is the set {2,7}, but the sets {1, 4} or {2,4,5} are not.

Determine a general formula for the number
of autonomous subsets of {1, 2, 3, ... , n}.

  Submitted by Ady TZIDON    
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Solution: (Hide)
The subset of k numbers must consist of the number k itself + any subset of n-1 members other than k, i.e. one of C(n-1, k-1) possible combinations. Summing for all possible values of k (1 to n-1):
Sum=C(n-1,0)+ C(n-1,1)+ C(n-1,2)+... C(n-1, n-1) =2^(n-1).

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): SolutionAdy TZIDON2014-07-30 17:00:49
re: SolutionCharlie2014-07-30 09:57:26
SolutionSolutiontomarken2014-07-30 08:48:04
Hints/Tipsre: computer assisted solutionAdy TZIDON2014-07-29 23:53:21
Solutioncomputer assisted solutionCharlie2014-07-29 14:38:40
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